
\(A_1^\)  is partially non-resonant,
\(\forall r\in R,\;\forall i,j\in I_r,\quad A_1^{}_{ii}-A_1^{}_{jj}\notin \lbrace 0\rbrace ,\) 
there exists a \(z\) -formal base change of the form \(+zP_1+\cdots \)  which transforms the matrix of \(\nabla ^o\)
\(A(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1\Big )\frac{z}{z}\) 
to the diagonal matrix
\(A^(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1^\Big )\frac{z}{z}.\) 
The result is standard ([1]}), but we will give a proof valid on any algebraic closed field of characteristic zero instead of \(\) , not depending on the transcendental notion of fundamental solution. Firstly, on can find a formal base change \(F_o=-zF_1^(t_o)+\cdots \)  which transforms \(A(t_o,z)\)  to the formal matrix
\(A(t_o,z)=\Big (\frac{\Lambda (t_o)}{z}+A_1^+zA_2+\cdots \Big )\frac{z}{z},\) 
where \(A_2,\dots \)  are block-diagonal with respect to \(r\in R\) , so that \(A(t_o,z)\)  is also block-diagonal. Each block (\(r\in R\) ) is written as
\(A^{(r)}(t_o,z)=\Big (\frac{t_{o,r}^{(r)}}{z}+A_1^{(r)}+zA^{(r)}_2+\cdots \Big )\frac{z}{z}.\) 
Since \(A_1^{(r)}\)  is non-resonant, there exists a base change \(^{(r)}+zQ^{(r)}_1+\cdots \)  which transforms \(A^{(r)}(t_o,z)\)  to the \(r\) th block of \(A^(t_o,z)\) .
Let \(t_o\in \Delta \)  and let \(U\)  be a connected open subset of \(T\)  satisfying (REF )–(REF ) with respect to \(S(t_o)\) . Under Assumptions REF (REF ) and (REF ) on \(A_1\) , there exists a holomorphic hypersurface \(\Theta \)  in \(U\)  not containing \(t_o\)  and a holomorphic matrix \(F_1^(t)\)  on \(U\Theta \) , such that the meromorphic connection on the trivial bundle with matrix (REF ) is integrable, restricts to (REF ) at \(t_o\) , and is formally equivalent at \(z=0\)  to the matrix connection
\(-(\Lambda (t)/z)+A_1^\,\frac{z}{z}.\) 
The proof is very similar to that of [2]}, [3]}. We set \(T=\lbrace z=0\rbrace \subset X=^n\times _z\) . We denote by \(\)  the \(T\) -meromorphic flat bundle \(\bigoplus _{i=1}^n(^{-t_i/z}\otimes _i)\) , where \(_i\!=\!(_X(*T),+A_1^{}_{ii}\,z/z)\) . By Lemma REF , (REF ) defines an object \((^{t_o},_{0})\)  in \(_0(\gamma _{t_o}^+)\) . From Theorem  we deduce an \(\) -marked \(T\) -meromorphic flat bundle \((,_{T})\)  on \(U\times (_z,0)\) . We can now apply [4]} and obtain a hypersurface \(\Theta \subset U\)  and a basis \(\)  of \((*(\Theta \times _z))\)  in which the matrix of the integrable connection takes the form
\(\Big (\frac{A_0(t)}{z}+A_1\Big )\frac{z}{z}+\frac{C(t)}{z}\) 
with \(A_0(t)\)  conjugate to \(\Lambda (t)\)  for each \(t\in U\Theta \)  and \(C(t)\in \Gamma (U,\Omega ^1_T(*\Theta ))\) . One can then apply the results in [4]}, since the regularity property of \(\Lambda (t)\)  (the fact that the eigenvalues are pairwise distinct) is not needed at this point. The change of notation with respect to  is as follows: \(\Delta _0\)  is \(\Lambda (t)\) , \(\Delta _\infty \)  is \(A_1^\) , \(\tau \)  is \(z\)  and the non-diagonal part \(T^\)  is \(F_1^(t)\) .
Application to the construction of Frobenius manifolds
It is known ([6]}) that, given suitable initial data consisting of a diagonal matrix \(\Lambda (t_o)\)  with pairwise distinct eigenvalues (\(t_o\notin \Delta \) ), of a matrix \(A_1\)  such that \(A_1-(w/2)\)  is skew-symmetric for some integer \(w\) , and an eigenvector \(\omega _o\)  of \(A_1\)  which has no zero entry, one can construct a Frobenius manifold structure on the complement of some hypersurface \(\Theta \)  in the universal covering \({T\Delta }\)  (also [4]}). It is the universal model at a semisimple point of a Frobenius manifold.
Theorem REF  enables us to relax the regularity assumption on \(\Lambda (t_o)\) , provided that Assumptions REF (REF ) and (REF ) are fulfilled for \(A_1\) . Let \(t_o\in \Delta \)  and set \(S_o=S(t_o)\) . Assume that \((\Lambda (t_o),A_1,\omega _o)\)  satisfy the following properties.

\(A_1^\in (\Lambda (t_o))\) ,


\(A_1-(w/2)\)  is skew-symmetric (so that \(A_1^=(w/2)\)  is non-resonant),


\(\omega _o\)  is an eigenvector of \(A_1\)  whose entries are all nonzero.

Let \(U\)  be an open subset of \(T\)  containing \(S_o\)  and satisfying (REF ) and (REF ), and let \(U\)  be its universal covering. By Theorem REF , there exists an integrable deformation of the Birkhoff normal form (REF ), which exists on \(U\Theta \)  for some complex hypersurface \(\Theta \)  of \(U\) . The vector \(\omega _o\)  can be extended flatly as a vector function \(\omega \)  on \(U\) , meromorphic along \(\Theta \)  and its entries do not vanish away from some hypersurface \(\Theta _{\omega _o}\) . To \(\omega \)  is then associated an infinitesimal period mapping ([4]}).
Under Assumptions (REF ), (REF ) and (REF ) above, the infinitesimal period mapping associated with \(\omega \)  endows the manifold \(U(\Theta \cup \Theta _{\omega _o})\)  with a Frobenius structure.
